Dictionary: A   B   C   D   E   F   G   H   I   J   K   L   M   N   O   P   Q   R   S   T   U   V   W   X   Y   Z

First-order logic

language, logic
The language describing the truth of mathematical formulas. Formulas describe properties of terms and have a truth value. The following are atomic formulas:
True False p(t1,..tn) where t1,..,tn are terms and p is a predicate.
If F1, F2 and F3 are formulas and v is a variable then the following are compound formulas:
F1 ^ F2 conjunction – true if both F1 and F2 are true,
F1 V F2 disjunction – true if either or both are true,
F1 => F2 implication – true if F1 is false or F2 is true, F1 is the antecedent, F2 is the consequent (sometimes written with a thin arrow),
F1 <= F2 true if F1 is true or F2 is false, F1 == F2 true if F1 and F2 are both true or both false (normally written with a three line equivalence symbol) ~F1 negation - true if f1 is false (normally written as a dash '-' with a shorter vertical line hanging from its right hand end). For all v . F universal quantification - true if F is true for all values of v (normally written with an inverted A). Exists v . F existential quantification - true if there exists some value of v for which F is true. (Normally written with a reversed E). The operators ^ V => <= == ~ are called connectives. "For all" and "Exists" are quantifiers whose scope is F. A term is a mathematical expression involving numbers, operators, functions and variables. The "order" of a logic specifies what entities "For all" and "Exists" may quantify over. First-order logic can only quantify over sets of atomic propositions. (E.g. For all p . p => p). Second-order logic can quantify over functions on propositions, and higher-order logic can quantify over any type of entity. The sets over which quantifiers operate are usually implicit but can be deduced from well-formedness constraints.
In first-order logic quantifiers always range over ALL the elements of the domain of discourse. By contrast, second-order logic allows one to quantify over subsets.
[“The Realm of First-Order Logic”, Jon Barwise, Handbook of Mathematical Logic (Barwise, ed., North Holland, NYC, 1977)].


Read Also:

  • First-papers

    plural noun, Informal. 1. an official declaration of intention filed by a resident alien desiring to become a U.S. citizen: not required by law after 1952.

  • First-past-the-post

    noun 1. (modifier) of or relating to a voting system in which a candidate may be elected by a simple majority rather than an absolute majority Compare proportional representation

  • First-person

    noun 1. the grammatical person used by a speaker in statements referring to himself or herself or to a group including himself or herself, as I and we in English. 2. a form in the first person. noun 1. a grammatical category of pronouns and verbs used by the speaker to refer to or talk […]

  • First-person shooter

    noun 1. a type of video game in which the player assumes the field of vision of the protagonist, so that the game camera includes the character’s weapon, but the rest of the character model is not seen. Abbreviation: FPS. noun 1. a type of computer game in which the player aims and shoots at […]

Disclaimer: First-order logic definition / meaning should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional. All content on this website is for informational purposes only.